COMBINED ESPRIT-ROOTMUSIC FOR DOA-DOD ES-TIMATION IN POLARIMETRIC BISTATIC MIMO RADAR
M. L. Bencheikh and Y. Wang
IREENA Laboratory — SEC Team Polytech’Nantes, Nantes, France
Abstract—In this paper, we investigate the exploitation of the polarimetric diversity signal properties in a bistatic polarimetric MIMO radar to improve the performance of joint estimation of direction of arrival (DOA) and direction of departure (DOD) of targets using combined ESPRIT-RootMUSIC technique. Numerical simulations are carried out to illustrate the performance of the proposed approach.
1. INTRODUCTION
The Multi-Input Multi-Output (MIMO) radar uses multiple antennas to transmit simultaneously several linearly orthogonal waveforms. It also uses multiple antennas to receive the reflected signals [1].
A bistatic MIMO radar scheme has been considered in [2], where a two-dimensional (2-D) spatial spectrum estimation technique based on the Capon method is developed. The proposed method needs an exhaustive search through all the 2-D space to find the DOA and DOD of the targets.
Exploiting the same MIMO radar configuration, Jin et al. [3] have proposed a joint DOA and DOD estimation approach based on the ESPRIT technique. A closed form of the target DOA and DOD is obtained and automatically paired. However, the number of targets which can be localized by this method is smaller than the number of receivers.
With the same architecture of the bistatic MIMO radar, another ESPRIT-based method is proposed in [5] by exploiting the invariance
Received 3 February 2011, Accepted 23 March 2011, Scheduled 29 March 2011
property of both the transmitting array and receiving array to estimate the DOA and DOD of targets. The pairing is not obtained automatically by this algorithm, but an additional pairing operation is given.
In [7], we have proposed an approach based on the combination of both the ESPRIT and Root-MUSIC methods to jointly estimate DOA-DOD of targets for a bistatic MIMO radar. This approach allows an automatic pairing between the target’s DOA and DOD.
Polarization diversity have been proven to be useful in array signal processing and various types of radar systems [8, 9, 11, 12].
Recently, by exploiting the invariance property of both the transmitting array and the receiving array, a method based on the ESPRIT technique is proposed in [10], to jointly estimate the parameter DOA-DOD-polarization of target for a polarimetric bistatic MIMO radar, but a pairing operation is needed.
In this work, we propose an extension of the technique developed in [7] by exploiting the polarization diversity in reception antennas. Besides the polarisation diversity, this approach allows an automatic pairing between the target’s DOA and DOD. Simulation results are provided to illustrate the performance of the proposed approach.
2. SIGNAL MODEL
The considered system is a bistatic polarimetric MIMO radar system withM closely spaced transmit antennas andN closely spaced receive antennas. Both of them are uniform linear arrays (ULA) and all the elements are omnidirectional. ∆tand ∆rare the inter-element spacing at the transmitter and the receiver, respectively. It is assumed that the Doppler frequencies have almost no effect on the orthogonality of the signals. Therefore, the variation of the phase within pulses caused by Doppler frequency can be ignored. The targets range is assumed to be much larger than the aperture of transmit array and receive array. Each element of the transmitter transmits orthogonal waveforms rm, m = 1. . . M. These signals are reflected by a target
assumed at position (θr, θt) with θt and θr denoting DOD and DOA, respectively. For P targets located at the same range bin, the output of the entire matched filters at the receiver can be written as [1–3]
z(t) =Cs(t) +n(t) (1)
with thepth column of the mixing matrixCgiven by
c
³
θ(rp), θt(p)
´
=b
³
θ(tp)
´
⊗a
³
θ(rp)
´¯¯ ¯
where
s(t) =
s1(t) .. . sP(t)
=
α1ej2πfd1t .. . αPej2πfdPt
(3)
a(θr) = h
1 ej2π4rsin(λθr) . . . ej2π(N−1)
4rsin(θr) λ
iT
∈CN×1 is the
re-ceiver steering vector; b(θt) = h
1ej2π4tsin(λθt) . . . ej2π(M−1) 4tsin(θt)
λ
iT
∈CM×1 is the transmitter steering vector; ⊗is the Kronecker opera-tor; α is the reflection coefficient depending on the target radar cross section; ej2πfdt is due to the target Doppler shift; n(t) is the noise vector, whose elements are assumed to be independent, zero mean and complex Gaussian distributed.
2.1. Polarimetric Model
In order to exploit the polarization diversity in reception, we assume that the receiver is composed of N dual polarization antennas. For a transverse electromagnetic (TEM) wave incident on the array, the simplified horizontal and vertical polarization components of the target reflected signal can be expressed as [8, 9, 12]
zpo(t) = ·
C(v) 0 0 C(h)
¸ ·
s(v)(t)
s(h)(t)
¸
+
·
n(v)(t)
n(h)(t)
¸
(4)
where v and h denote the vertical and horizontal polarizations, respectively, and s(v) = [s
1(t) cos(γ1), . . . , sP(t) cos(γP)]T, s(h) = £
s1(t) sin(γ1)ejξ1, . . . , sP(t) sin(γP)ejξP ¤T
are the targets signal vectors for polarizations vand h, respectively, the noise terms are assumed to be uncorrelated and complex Gaussian distributed with polarization angle γ ∈[0, π/2] and polarization phase differenceξ ∈[−π, π].
Assuming that the mixing matrix is polarization-independent, i.e.,
C(v)(θ
r, θt) = C(h)(θr, θt) = C(θr, θt), the extended signal model can
be written as
zpo(t) =Fs(t) + ·
n(v)(t)
n(h)(t)
¸
(5)
where F =
·
C 0 0 C
¸ ·
G(v)
G(h)
¸
, and G(v) and G(h) are diagonal
matrices with diagonal vectorsg(v) and g(h) given by
g(v) = [ cos(γ
1), . . . ,cos(γP) ]
g(h) = £ sin(γ
1)ejξ1, . . . ,sin(γP)ejξP
Then, the covariance matrix of the extended observation vector
zpo can be expressed as
Rp =FΓsFH +σ2I2M N (7)
whereΓs is the covariance matrix ofs(t) and σ2 is the noise power. The directional vector for a single targetfis given by
f(θr, θt) = ·
b(θt)⊗a(θr) cos(γ)
b(θt)⊗a(θr) sin(γ)ejξ ¸
(8)
3. POLARIMETRIC COMBINED ESPRIT-ROOTMUSIC
The MUSIC technique is based on the orthogonality between the noise subspace and the signal subspace. These subspaces can be obtained by the eigenvalues decomposition of the covariance matrix
Rp ∈ C2M N×2M N
Rp =UspΓspUspH +σ2UnpUHnp (9)
The noise subspace Unp is orthogonal to the actual target
directional vectors, i.e.,
f(θr, θt)HUnpUnpHf(θr, θt) = 0, θr=θ
(p)
r
θt=θ(tp) ¯ ¯ ¯ ¯ ¯
p=1,...,P
(10)
wheref can be written as
f(θr, θt) = B(θt)
·
a(θr) cos(γ)
a(θr) sin(γ)ejξ ¸
B(θt) = ·
[b(θt)⊗IN] 0
0 [b(θt)⊗IN] ¸
This expression allows to estimate separately the directions. Therefore, the proposed approach can be subdivided in two steps. In the first step, we determine only the DOD by the ESPRIT method and in the second step, for each estimated DOD, the estimated direction matrix B(ˆθt) is substituted in (10) and the RootMUSIC technique is
applied to estimate the DOA.
According to the principle of ESPRIT technique [5], to exploit the rotational invariance, in the transmitter array, we define the submatrices Ct1 ∈ C(M N−N)×P and Ct2 ∈ C(M N−N)×P, related to the DOD, as
Ct1=
h
b1
³
θ(1)t
´
⊗a
³
θ(1)r
´
, . . . ,b1
³
θ(tP)
´
⊗a
³
θ(rP)
´i
Ct2=
h
b2
³
θ(1)t
´
⊗a
³
θ(1)r
´
, . . . ,b2
³
θ(tP)
´
⊗a
³
θ(rP)
withb1(θ(tp)) andb2(θ(tp)) are the first and the lastM−1 elements of
b(θt(p)), respectively.
Then, it is easy to show
Ft2 =Ft1Φt (12)
whereFt1(2)=
·
Ct1(2) 0 0 Ct1(2)
¸ ·
G(v) G(h)
¸
andΦtis a diagonal matrix
with its main diagonal elements given by ρ(tp) = ej2π4tsin(θ (p) t )
λ for
p= 1, . . . , P.
The vectors of Usp span the signal subspace, then there exists a
non singular transformation matrix Tsatisfying
Usp =FT (13)
Let Uspt1 and Uspt2 be the submatrices formed from Usp in the same way as that Ft1 and Ft2 are formed from F, we can write
Uspt1 =Ft1T, Uspt2 =Ft2T (14) Then, the diagonal elements ofΦtare the eigenvaluesωp=1,...,P of
the following matrix Ωt =T−1Φ
tT which satisfies Uspt2 = Uspt1Ωt, therefore, the DOD can be estimated by
ˆ
θ(tp)= arcsin
µ
λ
2π∆targ (ωp) ¶
(15)
Therefore, for each estimated angle ˆθ(tp), ifθris the corresponding
direction of an actual target, then the directional vector is orthogonal to the noise subspace which allows to rewrite the expression (10) as
˙gHAT(1/zr) ·
Πn11 Πn12 Πn21 Πn22
¸
A(zr)˙g= 0 (16)
whereA(zr) = ·
a(zr) 0
0 a(zr) ¸
,zr=ej2π
4rsin(θr) λ , ˙g=
·
cos(γ) sin(γ)ejξ
¸
,
and
·
Πn11 Πn12
Πn21 Πn22
¸
=B(ˆθt)HU
npUHnpB(ˆθt).
The set (16) can be solved by finding the roots of the following polynomial
det
µ
AT(1/zr) ·
Πn11 Πn12
Πn21 Πn22
¸
A(zr) ¶
= 0 (17)
which can be expressed by 4XN−3
k=1
The parametersqk are calculated as
qk=
min[2XN−1,k]
i=max[1,k−2N+2]
Q(i, k−i+ 1) (19)
whereQ=v11vT22−v12vT21, and the elements ofvij are given by
vij(k) =
min[XN,N−k]
l=max[1,1−k]
[Πnij]l,k+l
¯ ¯ ¯ ¯ ¯ ¯
k=−(N−1),...,(N−1)
(20)
Therefore, the estimation of the angle θr which minimizes the
projection of the directional vector on the noise subspace, is equivalent to finding the roots of the polynomial (18) for each given ˆθt.
For each estimated DOD, the roots ˆzr inside and closest to
the unitary circle of the polynomial (18) allow to estimate the corresponding DOA angle.
ˆ
θ(rp) = arcsin
µ
λ
2π∆r arg (ˆzr) ¶
(21)
We note that, for this proposed method the pairing is automatically obtained between the DOD and DOA angles.
This approach exploiting the polarimetric diversity in bistatic polarimetric MIMO radar is able to handle more targets than the classical techniques. The maximum number of targets can be localized by this approach is 2N(M−1).
4. SIMULATION RESULTS
In this section, we investigate the performance of the proposed approach. For all the simulations, the parameters of targets ((α, fd, γ, ξ) are randomly generated for each target.
Firstly, we consider a bistatic polarimetric MIMO radar composed of M = 3 transmitter elements and N = 2 elements in receiver array spaced by a half wavelength. The data length isL= 500 samples and the number of trials is K = 200. For SN R = 20 dB, Figure 1 shows that the proposed approach is able to localize up toP = 8 targets.
Secondly, we set M = 3, N = 4, L = 256 and K = 200
for two targets (P = 2) located at the angles (θr, θt) = (35◦,60◦)
(θr, θt) = (80◦,15◦), Figures 2 and 3 show the Root Mean Square
0 10 20 30 40 50 60 70 80 90 0
10 20 30 40 50 60 70 80 90
DOA (°)
DOD (°
)
Figure 1. Joint DOD-DOA estimation of P = 8 targets by the proposed approach.
0 5 10 15 20 25 30 35 10-3
10-2 10-1 100 101 102
SNR (dB)
RMSE DOA (
°)
ESPRIT RootMUSIC Target 1 ESPRIT RootMUSIC Target 2 Polarimetric ESPRIT RootMUSIC Target 1 Polarimetric ESPRIT RootMUSIC Target 2 CRB Target 1
CRB Target 2
Figure 2. RMSE for DOA estimation versus SNR.
0 5 10 15 20 25 30 35 10-3
10-2 10-1 100 101
SNR (dB)
RMSE DOD (°
)
ESPRIT RootMUSIC Target 2 ESPRIT RootMUSIC Target 1 Polarimetric ESPRIT RootMUSIC Target 2 Polarimetric ESPRIT RootMUSIC Target 1 CRB Target 1
CRB Target 2
Figure 3. RMSE for DOD estimation versus SNR.
Therefore, the exploitation of the polarization diversity in reception improves considerably the estimation performance because the observations space is extended to 2M N × 2M N from M N ×
M N which increases also the number of localizable targets and the resolution power.
5. CONCLUSION
We have exploited the additional degrees-of-freedom of the polarization diversity to improve the performances of the combined ESPRIT-RootMUSIC proposed in [7] for joint DOA-DOD estimation in a bistatic MIMO radar. The exploitation of the polarimetric diversity allows to handle more targets than the classical algorithms and improve the localization performances.
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